Question

A and B bought lands on the Moon from an eStore, both with the same diameter but A’s land is square-shaped, and B’s land is circular. What is the ratio of the areas of their respective lands?

• A. 4 : π
• B. π : 4
• C. 1 : π
• D. π : 1

Hint:

For shapes with the same diameter, the square will always have a larger area than the circle because a square’s corners extend beyond the circle’s boundary. This holds for any diameter comparison.

Answer 1

The Correct Option is A

To find the ratio of the areas of the lands owned by A and B, we need to compare the area of a square with the area of a circle, both having the same diameter.

1. Let's assume the diameter of both the square and the circular lands is d.

2. For the square land owned by A:

   • The side of the square is equal to the diameter d.
   • Thus, the area of A's square land is calculated as follows:

   Area_{\text{square}} = d \times d = d^2

3. For the circular land owned by B:

   • The diameter of the circle is d, so the radius r is \frac{d}{2}.
   • Thus, the area of B's circular land is calculated as follows:

   Area_{\text{circle}} = \pi \times \left(\frac{d}{2}\right)^2 = \pi \times \frac{d^2}{4}

4. Now, let's calculate the ratio of the areas of A's square land to B's circular land:

   \text{Ratio} = \frac{Area_{\text{square}}}{Area_{\text{circle}}} = \frac{d^2}{\pi \times \frac{d^2}{4}} = \frac{d^2 \times 4}{\pi \times d^2} = \frac{4}{\pi}

Therefore, the ratio of the areas of A's and B's lands is 4 : \pi, making the correct option 4 : π.

Answer 2

The diameter of the square and the circle is the same. Let the diameter be \(d\).

Area of A's square land: A square's side length is equal to its diameter \(d\). Area of the square = \(d^2\).

Area of B's circular land: A circle's area is given by \(\pi r^2\), where \(r\) is the radius. Radius \(r = \frac{d}{2}\). Area of the circle = \(\pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\).

Ratio of Areas:

\[ \text{Ratio} = \frac{\text{Area of square}}{\text{Area of circle}} = \frac{d^2}{\frac{\pi d^2}{4}} = \frac{4}{\pi}. \]

Thus, the ratio of their areas is \(4 : \pi\).